I watched this video https://www.youtube.com/watch?v=DGpgt8j-nzw in which \( \int \ln ( \sqrt{x+1} + \sqrt{x} ) dx \) is computed by using trigonometry. Reading the comments, other solutions by using special functions are shown. However, the final antiderivatives do not use any of these functions, so these are intermediate steps in the most literal interpretation of that phrase. I thought of a solution that sidesteps the need of considering any functions other than those present in the integral: